THE NONLINEAR ACOUSTOELECTRIC EFFECT IN a CYLINDRICAL QUANTUM WIRE WITH AN INFINITE POTENTIAL

The nonlinear acoustoelectric effect in a cylindrical quantum wire with an infinite po-tential is investigated by using Boltzmann kinetic equation for an acoustic wave whose wavelength λ

THE NONLINEAR ACOUSTOELECTRIC EFFECT IN

A CYLINDRICAL QUANTUM WIRE WITH

AN INFINITE POTENTIAL

NGUYEN VAN NGHIA1,2, TRAN THI THU HUONG2

1Department of physics, Water Resources University

2Department of physics, Hanoi National University

NGUYEN QUANG BAU Department of physics, Hanoi National University

Abstract The nonlinear acoustoelectric effect in a cylindrical quantum wire with an infinite po-tential is investigated by using Boltzmann kinetic equation for an acoustic wave whose wavelength

λ = 2π

q is smaller than the mean free path l of the electrons and hypersound in the region ql  1, (where q is the acoustic wave number) The analytic expression for the acoustoelectric current I ac

is calculated in the case: relaxation time of momentum τ is constant approximation and degener-ates electrons gas The nonlinear dependence of the expression for the acoustoelectric current I ac

on the acoustic wave numbers q and on the intensity of constant electric field E are obtained Nu-merical computations are performed for AlGaAs/GaAs cylindrical quantum wire with an infinite potential The results are compared with the normal bulk semiconductors and the superlattices to show the values of the acoustoelectric current Iac in the cylindrical quantum wire are different than they are in the normal bulk semiconductors and the superlattices.

When an acoustic wave is absorbed by a conductor, the transfer of the momentum from the acoustic wave to the conduction electron may give rise to a current usually called the acoustoelectric current, Iac, in the case of an open circuit, a constant electric field The study of acoustoelectric effect in bulk materials have received a lot of attention [1-5] Recently, there have been a growing interest in observing this effect in mesoscopic structures [6-8] The interaction between surface acoustic wave (SAW) and mobile charges

in semiconductor layered structures and quantum wells is an important method to study the dynamic properties of low-dimensional systems The SAW method was applied to study the quantum Hall effects [9-11], the fractional quantum Hall effect [12], and the electron transport through a quantum point contact [13, 14] It has also been noted that the transverse acoustoelectric voltage (TAV) is sensitive to the mobility and to the carrier concentration in the semiconductor, thus it has been used to provide a characterization

of electric properties of semiconductors [15] Especially, in recent time the acoustoelectric effect was studied in both a one-dimensional channel [16] and in a finite-length ballistic quantum channel [17, 18, 19] In addition, the acoustoelectric effect was measured by an experiment in a submicron-separated quantum wire [20], in a carbon nanotube [21], in an InGaAs quantum well [22] The SAW method was also applied to the study acoustoelectric effect and acoustomagnetoelectric effect [23, 24, 25]

However, the acoustoelectric effect in the quantum wire still opens for studying, in this paper, we examine this effect in a cylindrical quantum wire with an infinite potential for the case of electron relaxation time is not dependent on the energy and degenerate electron gas Furthermore, we think the research of this effect may help us to understand the properties of quantum wire material We have obtained the acoustoelectric current

Iac in the cylindrical quantum wire The nonlinear dependence of the expression for the acoustoelectric current Iac on acoustic wave numbers q has been shown Numerical calculations are carried out with a specific AlGaAs/GaAs quantum wire to clarify our results

II ACOUSTOELECTRIC CURRENT

By using the classical Boltzmann kinetic equation method in [23, 24, 25], we cal-culated the acoustoelectric current in quantum wire The acoustic wave is considered a hypersould in the region ql  1 (l is the electron mean free path, q is the acoustic wave number) Under such circumstances, the acoustic wave can be interpreted as monochro-matic phonons having the 3D phonon distribution function N (~k), and this function can

be presented in the form [25]

N (~k) = (2π)

3

~ω~vs

where ~ = 1, ~k is the current phonon wave vector, φ is the sound flux density, ω~and vsare the frequency and the group velocity of sound wave with the wave vector ~q, respectively

It is assumed that the sound wave and the applied electric field ~E propagates along the axis of the quantum wire The problem was solved in the quasi-classical case, i.e., 2δ  τ−1, (τ is the relaxation time) The density of the acoustoelectric current can be written in the form [26]

jac= 2e (2π)3

Z

with

Uac =2πφ

ω~vs

{|G~p−~q,~p|2[f (ε~p−~q) − f (ε~)]δ(εp−~~ q− ε~+ ω~) + |G~p+~q,~p|2[f (ε~p+~q) − f (ε~)]δ(ε~p+~q− ε~− ω~)} (3) Here ~p is the electron momentum vector, f (ε~) is the distribution function, G~p−~q,~p

is the matrix element of the electron-phonon interaction and ψi (i = x, y, z) is the root of the kinetic equation given by [28]

e

c(V × H)

∂ψi

here Vi is the electron velocity, V is the average drift velocity of the moving charges and

W~{ } = (∂f /∂ε)−1W {(∂f /∂ε) } The operator cW is assumed to be Hermitian [26] In

the τ approximation, cW~= 1/τ Furthermore, τ = constant, we shall seek the solution of Eq.(4) as

Substituting Eq.(5) into Eq.(4) and solving by the method of iteration, we get for the zero and the first approximation Inserting into Eq.(2) and taking into account the fact that

|G~p, ~p0|2= |G~0 ,~ p|2 (6)

We obtain for the density of the acoustoelectric current the expression

jiac= − eφ

2π2vsω~

Z

|Gp+~~ q,~p|2[f (εp+~~ q) − f (ε~)]×

× [Vi(~p + ~q)τ − Vi(~p)τ ]δ(ε~p+~q− ε~− ω~)d3p−

2φτ2 2π2mcvsω~

Z

|G~p+~q,~p|2[f (ε~p+~q) − f (ε~)]×

× [(~V (~p + ~q) × ~H)i− (~V (~p) × ~H)i]δ(εp+~~ q− ε~− ω~)d3p (7) The matrix element of the electron-phonon interaction [23, 28] is given

|G~p,~q|2 = |Λ|

2|~q|2

Where Λ is the deformation potential constant and ρ is the crystal density of the quantum wire

In solving Eq.(7) we shall consider a situation whereby the sound is propagating along the quantum wire axis (Oz) Under such orientation the second term in Eq.(7)

is responsible for the density of the acoustomagnetoelectric current and the first term in Eq.(7) is the density of the acoustoelectric current Thus the density of the acoustoelectric current in Eq.(7) in the direction of the quantum wire axis becomes

jiac = −eφ~q

2τ |Λ|2 4π2vsω2

~ρ

Z [f (ε~p+~q) − f (ε~)][Vz(~p + ~q) − Vz(~p)]δ(ε~p+~q− ε~− ω~)d3p, (9)

the distribution function f (ε~) in the presence of the applied constant field ~E is obtained

by solving the Boltzmann equation in the τ approximation This function is given

f (ε~) =

Z ∞

0

dt

τ exp(−

t

In the case degenerate electrons gas is given by

f0(ε~) = θ(εF − ε~) =



0 ε~ > εF

Where εF is the Fermi energy, the energy ε~ of the cylindrical quantum wire with an infinite potential in the lowest miniband is given by [27]

ε~= ~

2~2z 2m +

~2A2n,l

Where l = 1, 2, 3, is the radial quantum number, n = 0, ±1, ±2, is the azimuth quantum number, m is the electron effective mass, R is the radius of the quantum wire, pz

is the longitudinal (relative to the quantum wire axis) component of the quasi-momentum and An,l is the l level root of Bessel function of the order n

Hence

Vz(~p) = ∂ε~

∂p =

~2pz

~2 2mR2

∂A2n,l

Substituting Eqs.(11), (12) and (13) into Eq.(9), we obtain for the acoustoelectric current with the condition is satisfied then:

εF > ~

2~2z 2m +

~2

The inequalities in Eq.(14) is condition acoustic wave vector ~q to the acoustoelectric effect exists Therefor, we have obtained the expression density of the acoustoelectric current

jzac = eφτ |Λ|

2q3 4πρvsω~2

Z ∞

0

dt

τ exp(−

t

τ)

h

~q − 2eEt − 2

s

2mεF −~

2A2 n,l

R2

i

Thus, the analytic expression for the acoustoelectric current Iac in the cylindrical quantum wire with an infinite potential can be written in the form

Iac= eφτ |Λ|

2R2q3 4ρvsω~2

Z ∞

0

dt

τ exp(−

t

τ)

h

~q − 2eEt − 2

s 2mεF −~

2A2 n,l

R2

i

The Eq.(16) is the acoustoelectric current in the cylindrical quantum wire with an infinite potential in the case degenerate electron gas, the expression only obtained if the condition in Eq.(14) is satisfied

III NUMERICAL RESULTS

In this situation Eq.(16) was solved analytically and the result were given as

Iac = eφτ |Λ|

2R2q3 4ρvsω2

~

h

~q − 2eEτ − 2

s

2mεF −~

2A2n,l

R2

i

Eq.(17) is the acoustoelectric current in the cylindrical quantum wire with an infinite potential in the case degenerate electron gas The dependences of the expression for the acoustoelectric current Iac on the intensity of the electric field E, the frequency ω~ of the acoustic wave, the acoustic wave numbers q and the radius R of the quantum wire are obtained

In the paper, we consider a AlGaAs/GaAs cylinder quantum wire with an infinite potential The parameters used in the calculations are as follows [26, 28]: τ = 10−12s; R = 80˚A; φ = 1014W m−2; ρ = 2×1013kgm−3; vs= 5370ms−1; E = 106V m−1; ω~ = 1010s−1; m = 0.067me, me being the mass of free electron

0 2 4 6 8 10 12

x 10 6

−0.02

−0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

q (m −1 )

ac (µA)

Fig The dependence of the acoustoelectric current Iac on the acoustic wave numbers q.

Figure shows the dependence of the acoustoelectric current on the acoustic wave number q when the relaxation time of momentum τ is constant approximation and degen-erate electron gas The curve of the acoustoelectric current Iac decreases when the small value range of the acoustic wave number q and strongly increases when the large value range of the acoustic wave number q

IV CONCLUSION

In this paper, we have analytically investigated the possibility of the acoustoelectric effect in the cylindrical quantum wire with an infinite potential We have obtained ana-lytically expressions for the acoustoelectric effect in the cylindrical quantum wire with an infinite potential for the case degenerate electron gas The dependences of the expression for the acoustoelectric current Iac on the frequency ω~ of the acoustic wave, the acous-tic wave numbers q and the radius R of the quantum wire are obtained The result is different compared to those obtained in the normal bulk semiconductors [5], according to [5] in the case τ = constant the effect only exists if the electron gas is non-degenerate,

if the electron gas is degenerate, the effect is not appear, however, our result indicates that in the cylindrical quantum wire with an infinite potential the acoustoelectric effect exists both non-degenerate and degenerate electron gas when τ = constant Unlike the normal bulk semiconductors, in the cylindrical quantum wire with an infinite potential the acoustoelectric current Iac is nonlinear with the acoustic wave number q

We have numerically calculated and graphed expressing the dependence of the acous-toelectric current Iac on the acoustic wave number q are performed for AlGaAs/GaAs cylindrical quantum wire with an infinite potential The result shows that, the acous-toelectric effect exists when the acoustic wave vector ~q complies with specific conditions

in Eq.(14) which condition dependences on the frequency ω~ of the acoustic wave, Fermi energy, the mass of electron and the radius R of the quantum wire That is mean to have acoustoelectric current Iac, the acoustic phonons energy is high enough and satisfied in the some interval to impact much momentum to the conduction electrons The curve of the

acoustoelectric current Iac strongly decreases when the small value range of the acoustic wave number q and strongly increases when the large value range of the acoustic wave number q

ACKNOWLEDGMENT This research is completed with financial support from the Program of Basic Re-search in Natural Science-NAFOSTED (103.01.18.09) and QG.TD.10.02

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Received 10-10-2010